## A numerical Galerkin method for a parabolic control problem |

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### Contents

Notation and Preliminaries | 6 |

A Discrete Time Galerkin Method for Parabolic | 18 |

The Continuous Problem | 40 |

3 other sections not shown

### Common terms and phrases

abritary adjoint assume backward discretization Boundary Value Problems Bramble and Thomee Broyden's method c(tn characterization coefficent matrix compact operator constant corresponding solution cp e H1 defined by 6.3 derived desired result dimensional subspace Dirichlet boundary conditions discrete analog Discrete Problem discrete time Galerkin Duhamel's principle Error estimates finite dimensional Galerkin approximations GALERKIN METHOD given z e Hence we obtain Hilbert space HP(ft HP(Q Hq(fi implies the desired independent integer interpolation J. H. Bramble Kk,h Lemma Math method defined nk _ TQ Note numerical approximation operators R(t optimal control orthogonal orthonormal systems parabolic equations positive semidefinite problem 4.1 Proof quadratic functional R(TQ Section selfadjoint solution of 4.6 solve t e 0,TQ Theorem 2.1 thesis triangle inequality u(TQ unique solution v e H1 vQ ll zd vQ ll+ vQ-v w(TQ x e Sh z e L ft zd-z